Let `S ={X in R, 0 lt x lt 1 and 2 tan^(-1)((1-x)/(1+x)) =cos^(-1)((1-x)^2/(1+x)^2)}`. If `n(S)` denotes the number of elements is S then

Options

A. `n(S) = 2` and only one element in S is less than `1/2` .
B. ` n(S) = 1` and the element in S is less than `1/2.`
C. `n(S) = 1` and the element in S is more than `1/2 .`
D. `n(S)=0`

Answer

To solve the problem, we need to analyze the given equation:

\[ S = \{ x \in \mathbb{R} : 0 < x < 1 \text{ and } 2 \tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{(1-x)^2}{(1+x)^2}\right) \} \]

### Step 1: Rewrite the equation using trigonometric identities

Using the identity for the tangent of a difference, we can rewrite the left-hand side:

\[ 2 \tan^{-1}\left(\frac{1-x}{1+x}\right) = \tan^{-1}\left(\frac{2(1-x)}{1+x}\right) \]

The right-hand side can be simplified using the cosine double angle identity:

\[ \cos^{-1}\left(\frac{(1-x)^2}{(1+x)^2}\right) = \cos^{-1}\left(\cos(2\theta)\right) \quad \text{where } \theta = \tan^{-1}(x) \]

### Step 2: Set up the equation

From the above, we can set up the equation:

\[ \tan^{-1}\left(\frac{2(1-x)}{1+x}\right) = \frac{\pi}{2} - 2\theta \]

### Step 3: Solve for \( x \)

Now, we can solve for \( x \) by substituting \( x = \tan(\theta) \):

\[ \tan^{-1}\left(\frac{2(1-\tan(\theta))}{1+\tan(\theta)}\right) = \frac{\pi}{2} - 2\tan^{-1}(\tan(\theta)) \]

This simplifies to:

\[ \frac{2(1-\tan(\theta))}{1+\tan(\theta)} = \tan\left(\frac{\pi}{2} - 2\tan^{-1}(\tan(\theta))\right) \]

### Step 4: Find the values of \( x \)

From the equation, we can find the specific values of \( x \) that satisfy the condition \( 0 < x < 1 \). We can use the known value of \( \tan(\frac{\pi}{8}) \):

\[ x = \tan\left(\frac{\pi}{8}\right) = \sqrt{2} - 1 \]

### Step 5: Check the range of \( x \)

Now, we need to check if \( \sqrt{2} - 1 \) lies within the interval \( (0, 1) \):

\[ \sqrt{2} \approx 1.414 \implies \sqrt{2} - 1 \approx 0.414 \]

Since \( 0 < \sqrt{2} - 1 < 1 \), we conclude that there is one solution in the interval.

### Conclusion

Thus, the number of elements in the set \( S \) is:

\[ n(S) = 1 \]